# Application of ideal class group?

I know what the class group is but I could not come up with any setting where we need to know what the class group actually is.

Does anyone know of an example where we have two number fields with the same class number but different class group, and the difference in class group lets us only use one of them for a number theory problem?

I can think of one example where different class number matter, but the class group is much more subtle.

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How about Hilbert class field? – Jiangwei Xue May 15 '11 at 20:26

3. As Xue says in a comment, class field theory provides a good example where class group structure matters, e.g., the fields Q(sqrt(-14)) and Q(sqrt(-30)) both have class number 4, but the first has cyclic class group and the second has class group that's a product of two groups of order 2. This means the number of quadratic unramified extensions of the two fields is not the same, since nonisom. groups of order 4 have different numbers of subgroups (equivalently, quotient groups) of order 2. More generally, class field theory tells us that class groups are Galois groups, and surely you appreciate that Galois theory is a lot more than theorems about the sizes of Galois groups! Moreover, class field theory says that any finite abelian extension of a number field $K$ has Galois group over $K$ which is (naturally) isomorphic to a generalized ideal class group of $K$. Only the abelian unramified extensions of $K$ are related to the ideal class group itself; you need generalized ideal class groups of $K$ to access the ramified abelian extensions. The moral is that you should think about the ideal class group merely as the most basic example of generalized ideal class groups, which taken all together tell you what the group structure of the maximal abelian extension of $K$ looks like. In more fancy language, the generalized ideal class groups are replaced by the group of ideles, and in the geometric setting of global fields with positive characteristic, generalized ideal class groups are replaced by generalized Jacobians to describe finite abelian extensions.